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SUSY DARK MATTER
Achille Corsetti and Pran Nath Department of Physics, Northeastern University, Boston, MA 02115-5005, USA
Abstract Several new effects have been investigated in recent analyses of supersymmetric dark matter. These include the effects of the uncertainties of wimp velocity dis- tributions, of the uncertainties of quark densities, of large CP violating phases, of nonuniversalities of the soft SUSY breaking parameters at the unification scale and of coannihilation on supersymmetric dark matter. We review here some of these with emphasis on the effects of nonuniversalities of the gaugino masses at the unification scale on the neutralino-proton cross-section from scalar interactions. The review encompasses several models where gaugino mass nonuniversalities occur including SUGRA models and D brane models. One finds that gaugino mass nonuniversalities can increase the scalar cross-sections by as much as a factor of 10 and also signifi- cantly extend the allowed range of the neutralino mass consistent with constraints up to about 500 GeV. These results have important implications for the search for supersymmetric dark matter. 1 INTRODUCTION
Over the recent past there has been considerable experimental activity in the di- rect detection of dark matter 1, 2) and further progress is expected in the ongoing experiments 1, 2, 3) and new experiments that may come online in the future 4). At the same time there have been several theoretical developments which have shed light on the ambiguities and possible corrections that might be associated with the predictions on supersymmetric dark matter. These consist of the effects on the dark matter analyses of wimp velocity 5, 6, 7) and of the rotation of the galaxy 8),the effects of the uncertainties of quark densities 9, 10, 11) and the uncertainties of the SUSY parameters 12), effects of large CP violating phases 13, 14), effects of scalar nonuniversalities 15), effects of nonuniversalities of gaugino masses 11) and effects of coannihilation 16). In this paper we will discuss some of these briefly but mainly focus on the effects of nonuniversalities of the gaugino masses on dark matter. In the Minimal Supersymmetric Standard Model (MSSM) there are 32 supersym- metric particles and with R parity conservation the lowest mass supersymmetric particle (LSP) is absolutely stable. In many unified models, such as in the SUGRA models 17), one finds that the lightest neutralino is the LSP over most of the pa- rameter space of the model. Thus the lightest neutralino is a candidate for cold 2 dark matter. The quantity that constrains supersymmetric models is Ωχh where
Ωχ = ρχ/ρc where ρχ is the density of relic neutralinos at the current temperatures, 2 and ρc =3H0 /8πGN is the critical matter density, and h is the Hubble parameter H0 in units of 100 km/sMpc. The most recent measurements of h from the Hubble Space Telescope give 18) h =0.71 0.03 0.07 (1) ± ± 19) Similarly the most recent analyses of Ωm give
Ω =0.3 0.08 (2) m ± If we assume that the component of Ω in Ω is Ω 0.05 which appears reasonable, B m B ' then this leads to the result Ω h2 =0.126 0.043. Perhaps a more cautious choice χ ± of the range would be a 2σ range which gives ∼ 0.02 Ω h2 0.3(3) ≤ χ ≤ The quantity of interest theoretically is
3 3 1/2 2 11 Tχ Tγ Nf Ωχh = 2.48 10− (4) ∼ × Tγ 2.73 J(xf ) Here Tf is the freeze-out temperature, xf = kTf /mχ˜ where k is the Boltzman constant, N is the number of degrees of freedom at the time of the freeze-out, ( Tχ )3 f Tγ is the reheating factor, and J(xf )isgivenby
xf 2 J (xf )= dx συ (x)GeV − (5) Z0 h i where <σv>is the thermal average with σ the neutralino annihilation cross-section and v the neutralino relative velocity.
2 Detection of Milky Way wimps
Both direct and indirect methods are desirable and complementary for the detec- tion of Milky Way wimps. We shall focus here on the direct detection. In this case the fundamental detector is the quark and the relevant interactions are the su- pergravity neutralino-quark-squark interactions. The scattering of neutralinos from quarks contains squark poles in the s channel and the Z boson and the Higgs bo- son (h, H0,A0) poles in the t channel. Since the wimp scattering from quarks is occuring at rather low energies one may, to a good approximation, integrate on the intermediate squark, Z and Higgs poles to obtain a low energy effective Lagrangian which gives a four-Fermi interaction of the following form
=¯χγ γ χqγ¯ µ(AP + BP )q + Cχχm¯ qq¯ + Dχγ¯ χm qγ¯ q (6) Leff µ 5 L R q 5 q 5 The contribution of D is generally small and thus the scattering is effectively gov- erned by the terms A,B and C. Analysis of dark matter is affected by several factors. We discuss these briefly below.
2.1 Uncertainties in wimp density and velocity
Two of the quantities that control the detection of dark matter are the wimp mass density and the wimp velocity. Estimates of Milky Way wimp density lie in the 20) 3 range ρ =(0.2 0.7)GeV cm− and the event rates in the direct detection χ − depend directly on this density. A second important factor regarding wimps that enters in the dark matter analyses is the wimp velocity. One typically assumes a Maxwellian velocity distribution for the wimps and the current estimates for the rms wimp velocity give v = 270 km/s with, however, a significant uncertainty. Estimates for the uncertainty lie in the range of 24 km/s to 70 km/s 21). A reasonable ± ± estimate then is that the rms wimp velocity lies in the range 21)
v = 270 50 km/s (7) ± Analyses including the wimp velocity variations show that the detection rates can have a significant variation, i.e., a factor of 2-3 on either side of the central values 7).
2.2 Effects of uncertainties of quark densities
The scattering of neutralinos from quarks are dominated by the scalar interaction which is controlled by the term C in Eq.(6). The dominant part of the scattering thus arises from the scalar part of the χ p cross-section which is given by − 2 4µr p 2 p 2 σχp(scalar)= ( fi Ci + (1 fi ) Ca) (8) π i=Xu,d,s 27 − i=Xu,d,s a=Xc,b,t
Here µ is the reduced mass in the χ p system and f p (i=u,d,s quarks) are quark r − i densities inside the proton defined by
m f p =
p mu σπN fu = (1 + ξ) mu + md mp p md σπN fd = (1 ξ) mu + md − mp p ms σπN fs = (1 x) (10) mu + md − mp wherewehavedefined ¯ ¯ σ0
σ =48 9 MeV, x =0.74 0.25,ξ=0.132 0.035 (12) πN ± ± ± p With the above range of errors one finds that fi lie in the range
f p =0.021 0.004,fp =0.029 0.006,fp =0.21 0.12 (13) u ± d ± s ± p Of these the errors in fs generates the largest variations. A detailed analysis shows that the scalar cross-section can vary by a factor of 5 in either direction due to errors in the quark densities 11). 2.3 CP violation effects on dark matter
The soft SUSY breaking parameters that arise in supersymmetric theories after spontaneous breaking are in general complex with phases O(1) and can lead to large electric dipole moment of the electron and of the neutron in conflict with current experiment. Recently a cancellation mechanism was proposed as a possible solution to this problem 22, 23). With the cancellation mechanism the total EDM of the electron and of the neutron can be in conformity with data even with phases O(1) and sparticle masses which are relatively light. The presence of large CP phases can affect dark matter and other low energy phenomena. Effects of CP violating phases on dark matter have been investigated for some time 13) and more recently such analyses have been extended to determine the effects of large CP phases under the cancellation mechanism 14). One finds that the EDM constraints play a crucial role in these analyses. Thus in the absence of CP violating phases one finds that the χ p cross-section can change by orders of magnitude when plotted as a function − of θµ (the phase of µ). These effects are, however, significantly reduced when the constraints arising from the current experimental limits on the electron and on the neutron EDMs are imposed. After the imposition of the constraints the effects of CP violating phases are still quite significant in that the χ p cross-section can vary − by a factor of 2. Thus precision predictions of the χ p cross-section should take ∼ − account of the CP phases if indeed such phases do exist in a given model. Indeed many string and D brane models do indeed possess such phases and thus inclusion of such phases is imperative in making predictions for direct detection in such models.
2.4 Effects of coannihilation
The effects of coannihilation may become important when the next to the lowest supersymmetric particle (NLSP) has a mass which lies close to the LSP mass 24).
∆ix The size of the effects is exponentially damped by the factor e− where ∆i = (m /m 1), x = m /kT and where m is the LSP mass. Because of this damping i χ − χ χ the coannihilation effects are typically important only for regions of the parameter space where the constraint ∆i < 0.1 is satisfied. Some of the possible candidates for 0 NLSP are the light stauτ ˜1,˜eR, the next to the lightest neutralino χ2, and the light + chargino χ1 . An interesting result one finds is that in mSUGRA the upper limit on the neutralino mass consistent with the current experimental constraints on the relic density is extended from 200 GeV to 600 GeV 16) when the effects of χ τ˜ − coannihilation are included. In Secs.(2.6) and (2.7) we will show that the allowed range of the neutralino mass can also be extended by inclusion of nonuniversalities in the gaugino masses.
2.5 Nonuniversality of scalar masses
The minimal SUGRA model is based on the universality at the GUT scale. This includes the universality of the scalar masses, of the gaugino masses and of the tri- linear couplings at the GUT scale. In supergravity unified models the universality of the soft SUSY breaking parameters arises from the assumption of a flat Kahler potential. However, the nature of physics at the GUT/Planck scale is not fully un- derstood and a more general analysis of the soft SUSY breaking sector requires that one work with a curved Kahler potential 25, 15). Such an analysis in general leads to nonuniversalities in the scalar sector of the theory. However, the nonuniversali- ties in the scalar sector cannot be completely arbitrary as there are very stringent constraints on the system from the limits on the flavor changing neutral currents (FCNC). A satisfaction of the constraints requires essentially a degeneracy in the scalar masses in the first two generations at the GUT scale. However, the constraints on the scalar masses in the Higgs sector and on masses in the third generation are far less severe and one could introduce significant amounts of nonuniversalities in these sectors without violating the FCNC constraints. It is found convenient to parametrize the nonuniversalities in the Higgs sector by δ1,δ2 so that at the GUT scale (M ) one has m2 = m2(1 + δ ), m2 = m2(1 + δ ). Similarly one may G H1 0 1 H2 0 2 parametrize the nonuniversalities in the third generation squark sector by δ3,δ4 so 2 2 2 2 that at the scale MG one has m = m (1+δ3), m = m (1+δ4). These nonuni- Q˜L 0 U˜R 0 versalities have a significant effect on the low energy physics. One of the main effects that occurs is through the effect on µ which is determined via the constraint of the radiative breaking of the electro-weak symmetry and is modified in the presence of the nonuniversalities in the Higgs sector and in the third generation sector. To one loop order it is given by 15)
m2 D 1 µ2 = µ2 + 0 (δ δ t2 0 − (δ + δ + δ )t2)+∆µ2 (14) 0 t2 1 1 − 2 − 2 2 3 4 − Here µ0 is the value of µ in the absence of nonuniversalities, D0 depends on the top Yukawa coupling and defines the position of the Landau pole, t tanβ,and ≡ ∆µ2 is the loop correction. We note that the entire effect of nonuniversalities is now explicity exhibited. One finds that the universalities can significantly affect the event rates. The effect on the event rates occurs specifically because of the effect on µ. Thus one finds that for certain regions of the parameter space the nonuniversalities in the Higgs and in the third generation sector make a negative contribution to µ2 which leads to larger higgsino components for the neutralino. Since in the direct detection the scattering is dominated by the scalar χ p cross-section which in turn − depends on the product of the gaugino and the higgsino components one finds that a smaller µ leads to larger event rates in the direct detection. A detailed analysis of the effects of nonuniversalities of the scalar masses has been given in Refs. 15). We will discuss further this phenomena in the context of the nonuniversalities in the gaugino sector in the next section.
2.6 Gaugino nonuniversalities and dark matter in GUT models
Nonuniversality of gaugino masses arises in grand unified models via corrections to the gauge kinetic energy functions 26). Thus in grand unified models a non-trivial gauge kinetic energy function leads to a gaugino mass matrix which has the form 26)
1 G/2 a 1 b b 1 m = e¯ G (G− ) (∂f∗ /∂z∗ )f − (15) αβ 4 a αγ γβ As an example if we consider the GUT group to be SU(5) then the gauge kinetic energy function fαβ transforms as follows (24 24) = 1 + 24 + 75 + 200 (16) × symm where (24 24) stands for the symmetric product. The term that transforms × symm like the singlet of SU(5) in the gauge kinetic energy function leads to universality of the gaugino masses, while the 24 plet, the 75 plet and the 120 plet will generate corrections to universality. In general one could have an admixture of the various representations and this will lead to gaugino masses of the form
r m˜ i(0) = m 1 (1 + crn ) (17) 2 i Xr r where ni depend on r and for the representations 1, 24, 75, 210 they are given in Table 1 27). The nonuniversality of the gaugino masses also leads to corrections of the gauge coupling constants at the GUT scale and in general one has gi(MG)= r gG(1 + r cr0 ni ). We note, however, that the coefficients cr0 that enter in gi are P different than those that enter in mi. This is so because the corrections to gi involve only the gauge kinetic energy function while the corrections to mi involve the gauge kinetic energy function as well as the nature of GUT physics.
Table: nonuniversalities at MX . r r r SU(5) rep n1 n2 n3 1 1 1 1 24 -1 3 2 75 -5 −3 1 200 10 2 1 The gaugino sector nonuniversalities affect µ. To exhibit this effect we can expand µ determined via the constraint of the radiative breaking of the electro-weak symmetry in terms of the parameter cr. One finds the following expansion
2 2 2 ∂µ˜ 2 µ˜ = µ0 + cr + O(cr) (18) Xr ∂cr 11) and for c24 < 0,c75 < 0,c200 > 0 one has ∂µ2 ∂µ2 ∂µ2 24 > 0, 75 > 0, 200 < 0 (19) ∂c24 ∂c75 ∂c200 Thus in these cases the nonuniversalities lead to a smaller value of µ .Nowas | | already mentioned in the previous section the Higgsino components become more dominant as µ becomes smaller. We can exhibit this analytically for the case when 28) 2 2 µ is small but we are still in the scaling region where µ /MZ >> 1. In this case it is possible to analytically investigate the size of the gaugino-Higgsino components
Xn0 of the LSP defined by ˜ ˜ ˜ ˜ χ = X10B + X20W3 + X30H1 + X40H2 (20) ˜ ˜ ˜ ˜ where B is the Bino, W3 is the Wino, and H1,andH2 are the two Higgsinos. In this case one finds that the gaugino components of the LSP are given by X11 2 2 ' MZ 2 MZ 1 ( 2µ2 )sin θW ,andX12 2m2 µ sin2θW sinβ while the higgsino components are − ' χ1 given by 28) X MZ sin2θ sinβ, X MZ sin2θ sinβ. From the above one 13 '− µ W 14 ' µ W finds that the Higgsino components have a dependence on the inverse power of µ and thus a smaller µ will lead to a larger scalar σχ p cross-section. The literature on the − analyses of dark matter relic density 29, 30) and direct detection in MSSM 31) and in SUGRA models 32) is quite extensive 33). We discuss here the quantitative effects of the gaugino mass nonuniversality on dark matter. Some features of the effects of gaugino mass nonuniversalities have already been discussed in the literature 34) and we review here the more recent developments 11). The techniques used in the analysis are as discussed in Ref. 32) and in the analysis we impose the b s + γ → constraint 35). In Fig.1 we plot the scalar χ p cross-section as a function of the − neutralino mass for the case of GUT scale nonuniversalities with values of c24 in the range -0.1 to 0.08. One finds that the scalar cross-section is enhanced for negative values of c24 just as one would expect from the general discussion above because it is for the case of c24 negative that µ becomes small. One finds that in general the scalar cross-section increases systematically as c increases for negative values | 24| of c24 and an enhancement of the scalar cross-section by as much as a factor of 10 6e−43
1 + 24 c =−0.1 5e−43 24
4e−43 ) 2
c =−0.08 3e−43 24 (scalar) (cm p χ σ 2e−43
c24=−0.04
c =−0.06 1e−43 24
c24=−0.02 c24=0.08
c24=0.06 c24=0 c24=0.04 c24=0.02 0e+00 50 60 70 80 90 100 110 120
mχ (GeV)
Figure 1: σ (scalar) vs m when m =51GeV, tan β =10, A /m = 7 and c χp χ 0 t 0 − 24 takes on various values (Taken from Ref. 11)).
can be gotten relative to the universal case of c24 = 0. One also finds an enhance- ment of the allowed range of the neutralino mass consistent with the constraints. In Fig.2 we plot the maximum and the minimum of the scalar cross-section as a function of the neutralino mass for the case of GUT scale nonuniversalities where the nonuniversalities arise from the 200 plet representation with c200 =0.1when the other parameters are varied over their assumed naturalness range. The current experimental limits from DAMA 1) and from CDMS 2) are also plotted. Further, the currents limits would certainly be significantly improved in other dark matter detectors in the future 2, 3, 4) and in Fig.2 we also plot the expected limits from future CDMS, and from GENIUS 4). One finds that the current experiment does constrain the theory in a small region of the parameter space. Further, the expected sensitivity in future experiment, i.e., in CDMS and in GENIUS will explore a ma- jor part of the parameter space of this model. We also note that the inclusion of nonuniversality significantly increases the allowed range of the neutralino parameter space.
2.7 Dark Matter on D Branes
Nonuniversality of gaugino masses is rather generic in string theory. However, the specific nature of the nonuniversality will depend on the details of the compactifi- 10−40
10−41
10−42 ) 2 10−43
−44 c =0.1 10 200 (scalar) (cm p χ σ
10−45
DAMA 10−46 CDMS CDMS (future) GENIUS 10−47 20 60 100 140 180 220 260 300 340 380
mχ (GeV)
Figure 2: The maximum and the minimum curves of σχp(scalar) vs mχ when c200 =0.1 and SUSY parameters are varied over the natualness domain (Taken from Ref. 11)). cation. We discuss here the effects of gaugino mass nonuniversality on dark matter in the context of D brane models. The possibility of nonuniversal gaugino phases in brane models arises from the choice of embedding of the different gauge groups of the Standard Model on different branes. One may consider, from example, models that arise from Type IIB string compactified on a six-torus T 2 T 2 T 2 which contains × × 9branes,7i and 5i (i=1,2,3) branes and 3 branes. Not all the branes can be present simultaneuosly due to the constraint of N=1 supersymmetry which requires that one has either 9 branes and 5i (i=1,2,3) branes or 7i (i=1,2,3) branes and 3 branes. In the following we will make the choice of embedding on 9 branes and 5 branes 23). One of the major problems in developing a sensible string phenomenology is that the mechanism of supersymmetry breaking in string theory is still lacking. However, some progress can be made by use of an efficient parametrization of supersymmetry breaking. Here we use the parametrization where the breaking of supersymmetry arises from the breaking generated by the dilaton and the moduli VEV’s of the fol-
36) S iγS i iγi lowing form F =√3m3/2(S + S∗)sinθe− , F =√3m3/2(Ti + Ti∗)cosθie− . We consider now a specific 9-5 brane model. Here one embeds the SU(3) U(1) C × Y gauge group on 9 branes and SU(2)L gauge group on a 51 brane. The alternative possibility of embedding the Standard Model gauge group on five branes is discussed in the last two papers of Ref. 22).Forthe9 5 brane model the soft SUSY breaking − 1 10−39
10−40 9−51 D Brane model
10−41
10−42 ) 2 10−43
10−44 (scalar) (cm p χ σ 10−45
10−46 DAMA CDMS −47 10 CDMS (future) GENIUS 10−48 0 100 200 300 400 500
mχ (GeV)
Figure 3: The maximum and the minimum curves of σχp(scalar) vs mχ for the 9-5 D brane model when c200 =0.1 when m3/2 ranges up to 2 TeV, tan β ranges up to 25, and θ lies in the range 0.1-1.6. (Taken from Ref. 11)). sector of the theory is given by 36, 23)
iγ iγ m˜ =˜m = √3m sinθe− S = A ;˜m = √3m cosθe− i 1 3 3/2 − 0 2 3/2 m˜ 2 = m2 (1 3cos2θΘ2);m ˜ 2 = m2 (1 (3/2)cos2θ(1 Θ2)) (21) 9 3/2 − 1 951 3/2 − − 1
Here θ(Θi) is the Goldstino direction in the dilaton S (moduli Ti) VEV space. We discuss now dark matter on D branes. In Fig.3 we give a plot of the scalar χ p cross − section as a function of the neutralino mass for the 9 5 D brane model. One of − 1 the interesting feature of the D brane model is that the scalar masses are in general not universal. However for Θ1 =1/√3 one has m9 = m951 and the scalar masses are universal although the gaugino masses are still nonuniversal. Since we are mostly interested here in investigating the effects of nonuniversalities of the gaugino masses in this analysis, we impose universality of the scalar masses and set Θ1 =1/√3. In Fig.3 we give a plot of the minimum and the maximum of the scalar χ p cross − section under this constraint. One finds that under the assumed constraints the allowed domain of the parameter space has the general features which are similar to the GUT scale nonuniversalities. One common feature is that the allowed domain of the parameter space is extended close to 500 GeV. One may note that if in addition to the constraint Θ1 =1/√3 one also sets θ = π/6 one finds also universality of the gaugino masses. This situation is exhibited by the vertical dark line in the enclosed region on the left hand side in Fig.3.
3 Conclusion
In this paper we have given a brief review of the recent theoretical developments in the analyses of supersymmeteric dark matter. Our emphasis has been in exploring the effects of uncertainties of the input data and the effects of nonuniversalities of the gaugino masses on dark matter analyses. It is found that the uncertainties of the wimp velocities can change detection rates by up to factors of 2-3 while the uncertainties in quark masses and densities can change the χ p cross-section by − up to factors of 5 in either direction. The effects of gaugino mass nonuniversalities on dark matter analyses is found to be quite dramatic. It is seen that gaugino mass nonuniversalities can increase the Higgsino components of the LSP and significantly increase the χ p cross-section from scalar interactions and also increase the allowed − range of the LSP consistent with relic density constraints. Thus an increase in the scalar χ p cross-sections by up to a factor of 10 can occur while the allowed − range of the neutralino masses can move up to 500 GeV consistent with the relic density constraints. Data from current dark matter experiments is beginning to put constraints on models with nonuniversalities. These constraints will become more severe as the sensitivity of dark matter experiments increase in the future. Acknowledgements This research was supported in part by NSF grant PHY-9901057.
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